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    Groups of order 18

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    Order $18$ has exactly five group types: two abelian and three nonabelian.Suggest a slogan

    Up to isomorphism there are five groups of order \(18\). The abelian ones are \(\ZZ/18\ZZ\) and \((\ZZ/3\ZZ)^2\times\ZZ/2\ZZ\), and the nonabelian ones are

    1. the dihedral group \(D_9\) of order \(18\),

    2. \(S_3\times\ZZ/3\ZZ\),

    3. the generalized dihedral group \((\ZZ/3\ZZ)^2\rtimes\ZZ/2\ZZ\), with the generator of \(\ZZ/2\ZZ\) acting by inversion.

    What are the groups of order 18?

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